<p>Let \(A(2, -3)\) and \(B(-2, 1)\) be vertices of a triangle \(ABC\). If the centroid of this triangle moves on the line \(2x + 3y = 1\), then the locus of the vertex <i>C</i> is the line</p>
Step-by-Step Solution
Key Concept: The centroid G of triangle ABC with vertices A(2,-3), B(-2,1), and C(h,k) is given by G = ((2-2+h)/3, (-3+1+k)/3). Since G lies on 2x+3y=1, substitute the centroid coordinates into this equation to find the locus of C.
<p><strong>Step 1:</strong> Use centroid formula. If C = (h, k), then centroid G = ((2-2+h)/3, (-3+1+k)/3) = ((h)/3, (k-2)/3)</p><p><strong>Step 2:</strong> Since centroid G moves on line 2x + 3y = 1, substitute G's coordinates:</p><p>2(h/3) + 3((k-2)/3) = 1</p><p><strong>Step 3:</strong> Simplify:</p><p>2h/3 + (k-2) = 1</p><p>2h/3 + k - 2 = 1</p><p>2h/3 + k = 3</p><p>2h + 3k = 9</p><p><strong>Step 4:</strong> Replace (h,k) with (x,y) for the locus:</p><p>∴ Answer: <strong>2x + 3y = 9</strong> (Option A)</p>
Correct Answer: A